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Full text of "Treatise On Analysis Vol-Ii"

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12 QUOTIENT GROUPS 57 There is also a canonical bijection if/ of the quotient group A/(A n H) onto the subgroup n(A) of G/H; this bijection \j/ is the unique mapping which makes the following diagram commutative: A (12.12.4) „,/ A/(AnH)-^ 7t(A) where n' is the canonical mapping and 7 : A ->G the canonical injection. It follows from (12.10.6) that \// is continuous, but it is not necessarily bicon- tinuous (Problem 2). However: (12.12.5) Let G be a metrizable group and H a compact subgroup ofG. For each closed subgroup A of G, the canonical bijection \//: A/(A n H) -> 7t(A) is an isomorphism of topological groups. It is enough to show that the image under \f/ of an arbitrary closed subset F of A/(A n H) is closed in 7i(A). Now ^"'(F) = B is a closed subset of A, hence closed in G, and ^(F) = n(j(B)) = 7i(BH). But since H is compact, the set BH is closed in G (12.10.5) and saturated, hence 7r(BH) is closed in G/H (12.10) and therefore also closed in 7r(A). (12.12.6) Let Gj, G2 be two topological groups and Hi (resp. H2) a normal subgroup ofGl (resp. G2). Then the canonical bijection , x H2) -Kd/H,) x (G2/H2) is an isomorphism of topological groups. This is an immediate consequence of (12.10.11). (12.12.7) Let G, G' be two topological groups and u: G ~»G' a continuous homorphism. Then u factorizes canonically as G 4> G/N 4- M(G) -i G' where N is the kernel of w,/? is the canonical homomorphism, and j the canon- ical injection. The mapping v is a continuous bijection (12.10.6(iii)), but is not necessarily bicontinuous. When v is bicontinuous, the homomorphism u is said to be a .yfnttf morphism of G into G'; for this to be the case, it is necessary and sufficient that for each neighborhood V of e in G, the set u(V) should be a neighborhood of e' = u(e) in u(G) (12.11.5).plied to the right action